Heterogeneous Relation Algebra
So far, relational algebra has been presented in its classical form. Relations are often conceived as something that might be called quadratic or homogeneous; a relation over a set. It is interpreted as a subset R ⊂ U × U of a Cartesian product of the universe U with itself. If relations between two or more sets are considered, this may easily be subsumed under this view, uniting all the sets in question into one huge set and calling this set the universe U. On the other hand, a variant of the theory has evolved that treats relations from the very beginning as heterogeneous or rectangular, i.e. as relations where the normal case is that they are relations between two different sets. The present chapter is devoted to this variant form.
KeywordsHomogeneous Case Relation Algebra Bijective Mapping Surjective Homomorphism Partial Identity
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- 1.To avoid set-theoretic problems we restrict ourselves to categories where every class of mor-phisms MorR [A, B] is a set. Such categories are called locally small.Google Scholar
- 2.Recall that in Chapt. 1 the small circle was used for composition of morphisms in preference to the semicolon. Here, however, we wish to emphasize that morphisms are to be thought of as relations, therefore we revert to the semicolon.Google Scholar
- 3.The universal covering got its name from a similar concept in the theory of Riemann surfaces.Google Scholar